Theorem mdandyvrx1 46999

Description: Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)

Hypotheses

Ref	Expression
mdandyvrx1.1	⊢ (𝜑 ⊻ 𝜁)
mdandyvrx1.2	⊢ (𝜓 ⊻ 𝜎)
mdandyvrx1.3	⊢ (𝜒 ↔ 𝜓)
mdandyvrx1.4	⊢ (𝜃 ↔ 𝜑)
mdandyvrx1.5	⊢ (𝜏 ↔ 𝜑)
mdandyvrx1.6	⊢ (𝜂 ↔ 𝜑)

Assertion

Ref	Expression
mdandyvrx1	⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜁))

Proof of Theorem mdandyvrx1

Step	Hyp	Ref	Expression
1		mdandyvrx1.2	. . . . 5 ⊢ (𝜓 ⊻ 𝜎)
2		mdandyvrx1.3	. . . . 5 ⊢ (𝜒 ↔ 𝜓)
3	1, 2	axorbciffatcxorb 46922	. . . 4 ⊢ (𝜒 ⊻ 𝜎)
4		mdandyvrx1.1	. . . . 5 ⊢ (𝜑 ⊻ 𝜁)
5		mdandyvrx1.4	. . . . 5 ⊢ (𝜃 ↔ 𝜑)
6	4, 5	axorbciffatcxorb 46922	. . . 4 ⊢ (𝜃 ⊻ 𝜁)
7	3, 6	pm3.2i 470	. . 3 ⊢ ((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜁))
8		mdandyvrx1.5	. . . 4 ⊢ (𝜏 ↔ 𝜑)
9	4, 8	axorbciffatcxorb 46922	. . 3 ⊢ (𝜏 ⊻ 𝜁)
10	7, 9	pm3.2i 470	. 2 ⊢ (((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜁))
11		mdandyvrx1.6	. . 3 ⊢ (𝜂 ↔ 𝜑)
12	4, 11	axorbciffatcxorb 46922	. 2 ⊢ (𝜂 ⊻ 𝜁)
13	10, 12	pm3.2i 470	1 ⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜁))

Syntax hints:   ↔ wb 206   ∧ wa 395   ⊻ wxo 1510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-xor 1511

This theorem is referenced by:  mdandyvrx14  47012